1/*- 2 * SPDX-License-Identifier: BSD-3-Clause 3 * 4 * Copyright (c) 1992, 1993 5 * The Regents of the University of California. All rights reserved. 6 * 7 * This software was developed by the Computer Systems Engineering group 8 * at Lawrence Berkeley Laboratory under DARPA contract BG 91-66 and 9 * contributed to Berkeley. 10 * 11 * Redistribution and use in source and binary forms, with or without 12 * modification, are permitted provided that the following conditions 13 * are met: 14 * 1. Redistributions of source code must retain the above copyright 15 * notice, this list of conditions and the following disclaimer. 16 * 2. Redistributions in binary form must reproduce the above copyright 17 * notice, this list of conditions and the following disclaimer in the 18 * documentation and/or other materials provided with the distribution. 19 * 3. Neither the name of the University nor the names of its contributors 20 * may be used to endorse or promote products derived from this software 21 * without specific prior written permission. 22 * 23 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 24 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 25 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 26 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 27 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 28 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 29 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 30 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 31 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 32 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 33 * SUCH DAMAGE. 34 */ 35 36#if defined(LIBC_SCCS) && !defined(lint) 37static char sccsid[] = "@(#)muldi3.c 8.1 (Berkeley) 6/4/93"; 38#endif /* LIBC_SCCS and not lint */ 39#include <sys/cdefs.h> 40__FBSDID("$FreeBSD$"); 41 42#include "quad.h" 43 44/* 45 * Multiply two quads. 46 * 47 * Our algorithm is based on the following. Split incoming quad values 48 * u and v (where u,v >= 0) into 49 * 50 * u = 2^n u1 * u0 (n = number of bits in `u_long', usu. 32) 51 * 52 * and 53 * 54 * v = 2^n v1 * v0 55 * 56 * Then 57 * 58 * uv = 2^2n u1 v1 + 2^n u1 v0 + 2^n v1 u0 + u0 v0 59 * = 2^2n u1 v1 + 2^n (u1 v0 + v1 u0) + u0 v0 60 * 61 * Now add 2^n u1 v1 to the first term and subtract it from the middle, 62 * and add 2^n u0 v0 to the last term and subtract it from the middle. 63 * This gives: 64 * 65 * uv = (2^2n + 2^n) (u1 v1) + 66 * (2^n) (u1 v0 - u1 v1 + u0 v1 - u0 v0) + 67 * (2^n + 1) (u0 v0) 68 * 69 * Factoring the middle a bit gives us: 70 * 71 * uv = (2^2n + 2^n) (u1 v1) + [u1v1 = high] 72 * (2^n) (u1 - u0) (v0 - v1) + [(u1-u0)... = mid] 73 * (2^n + 1) (u0 v0) [u0v0 = low] 74 * 75 * The terms (u1 v1), (u1 - u0) (v0 - v1), and (u0 v0) can all be done 76 * in just half the precision of the original. (Note that either or both 77 * of (u1 - u0) or (v0 - v1) may be negative.) 78 * 79 * This algorithm is from Knuth vol. 2 (2nd ed), section 4.3.3, p. 278. 80 * 81 * Since C does not give us a `long * long = quad' operator, we split 82 * our input quads into two longs, then split the two longs into two 83 * shorts. We can then calculate `short * short = long' in native 84 * arithmetic. 85 * 86 * Our product should, strictly speaking, be a `long quad', with 128 87 * bits, but we are going to discard the upper 64. In other words, 88 * we are not interested in uv, but rather in (uv mod 2^2n). This 89 * makes some of the terms above vanish, and we get: 90 * 91 * (2^n)(high) + (2^n)(mid) + (2^n + 1)(low) 92 * 93 * or 94 * 95 * (2^n)(high + mid + low) + low 96 * 97 * Furthermore, `high' and `mid' can be computed mod 2^n, as any factor 98 * of 2^n in either one will also vanish. Only `low' need be computed 99 * mod 2^2n, and only because of the final term above. 100 */ 101static quad_t __lmulq(u_long, u_long); 102 103quad_t 104__muldi3(a, b) 105 quad_t a, b; 106{ 107 union uu u, v, low, prod; 108 u_long high, mid, udiff, vdiff; 109 int negall, negmid; 110#define u1 u.ul[H] 111#define u0 u.ul[L] 112#define v1 v.ul[H] 113#define v0 v.ul[L] 114 115 /* 116 * Get u and v such that u, v >= 0. When this is finished, 117 * u1, u0, v1, and v0 will be directly accessible through the 118 * longword fields. 119 */ 120 if (a >= 0) 121 u.q = a, negall = 0; 122 else 123 u.q = -a, negall = 1; 124 if (b >= 0) 125 v.q = b; 126 else 127 v.q = -b, negall ^= 1; 128 129 if (u1 == 0 && v1 == 0) { 130 /* 131 * An (I hope) important optimization occurs when u1 and v1 132 * are both 0. This should be common since most numbers 133 * are small. Here the product is just u0*v0. 134 */ 135 prod.q = __lmulq(u0, v0); 136 } else { 137 /* 138 * Compute the three intermediate products, remembering 139 * whether the middle term is negative. We can discard 140 * any upper bits in high and mid, so we can use native 141 * u_long * u_long => u_long arithmetic. 142 */ 143 low.q = __lmulq(u0, v0); 144 145 if (u1 >= u0) 146 negmid = 0, udiff = u1 - u0; 147 else 148 negmid = 1, udiff = u0 - u1; 149 if (v0 >= v1) 150 vdiff = v0 - v1; 151 else 152 vdiff = v1 - v0, negmid ^= 1; 153 mid = udiff * vdiff; 154 155 high = u1 * v1; 156 157 /* 158 * Assemble the final product. 159 */ 160 prod.ul[H] = high + (negmid ? -mid : mid) + low.ul[L] + 161 low.ul[H]; 162 prod.ul[L] = low.ul[L]; 163 } 164 return (negall ? -prod.q : prod.q); 165#undef u1 166#undef u0 167#undef v1 168#undef v0 169} 170 171/* 172 * Multiply two 2N-bit longs to produce a 4N-bit quad, where N is half 173 * the number of bits in a long (whatever that is---the code below 174 * does not care as long as quad.h does its part of the bargain---but 175 * typically N==16). 176 * 177 * We use the same algorithm from Knuth, but this time the modulo refinement 178 * does not apply. On the other hand, since N is half the size of a long, 179 * we can get away with native multiplication---none of our input terms 180 * exceeds (ULONG_MAX >> 1). 181 * 182 * Note that, for u_long l, the quad-precision result 183 * 184 * l << N 185 * 186 * splits into high and low longs as HHALF(l) and LHUP(l) respectively. 187 */ 188static quad_t 189__lmulq(u_long u, u_long v) 190{ 191 u_long u1, u0, v1, v0, udiff, vdiff, high, mid, low; 192 u_long prodh, prodl, was; 193 union uu prod; 194 int neg; 195 196 u1 = HHALF(u); 197 u0 = LHALF(u); 198 v1 = HHALF(v); 199 v0 = LHALF(v); 200 201 low = u0 * v0; 202 203 /* This is the same small-number optimization as before. */ 204 if (u1 == 0 && v1 == 0) 205 return (low); 206 207 if (u1 >= u0) 208 udiff = u1 - u0, neg = 0; 209 else 210 udiff = u0 - u1, neg = 1; 211 if (v0 >= v1) 212 vdiff = v0 - v1; 213 else 214 vdiff = v1 - v0, neg ^= 1; 215 mid = udiff * vdiff; 216 217 high = u1 * v1; 218 219 /* prod = (high << 2N) + (high << N); */ 220 prodh = high + HHALF(high); 221 prodl = LHUP(high); 222 223 /* if (neg) prod -= mid << N; else prod += mid << N; */ 224 if (neg) { 225 was = prodl; 226 prodl -= LHUP(mid); 227 prodh -= HHALF(mid) + (prodl > was); 228 } else { 229 was = prodl; 230 prodl += LHUP(mid); 231 prodh += HHALF(mid) + (prodl < was); 232 } 233 234 /* prod += low << N */ 235 was = prodl; 236 prodl += LHUP(low); 237 prodh += HHALF(low) + (prodl < was); 238 /* ... + low; */ 239 if ((prodl += low) < low) 240 prodh++; 241 242 /* return 4N-bit product */ 243 prod.ul[H] = prodh; 244 prod.ul[L] = prodl; 245 return (prod.q); 246} 247